A387011 a(n) = Sum_{k=0..n} binomial(4*n+4,k).

1, 9, 79, 697, 6196, 55455, 499178, 4514873, 40999516, 373585604, 3414035527, 31278197839, 287191809724, 2642070371194, 24347999094724, 224723513577529, 2076978797223820, 19220104372823340, 178061257422521452, 1651314042800498052, 15328459501269535952

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

a(n) = [x^n] (1+x)^(4*n+4)/(1-x).

a(n) = [x^n] 1/((1-x)^(3*n+4) * (1-2*x)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+4,k) * binomial(4*n-k+3,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+3,n-k).

G.f.: g^5/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.

D-finite with recurrence: 128*(4*n-3)*(2*n+1)*(4*n-1)*(22*n^2+16*n-3)*a(n-2) -8*(1892*n^5+1024*n^4-1982*n^3-1306*n^2-60*n+27)*a(n-1) +3*(n+1)*(3*n+2)*(3*n+1)*(22*n^2-28*n+3)*a(n) = 0. - Georg Fischer, Aug 17 2025

a(n) ~ 2^(8*n + 15/2) / (sqrt(Pi*n) * 3^(3*n + 7/2)). - Vaclav Kotesovec, Aug 18 2025

a(n) = Sum_{k=0..floor(n/2)} binomial(4*n+5,n-2*k). - Seiichi Manyama, Nov 11 2025

Mathematica

Table[Sum[Binomial[4*n+4, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 16 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(4*n+4, k));
(Magma) [&+[Binomial(4*n+4, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025

Crossrefs

Cf. A066381, A386811, A387009, A387010, A387034, A387035, A387036, A387037.

Cf. A002293.

Sequence in context: A388414 A190979 A254598 * A083411 A181279 A173808

Adjacent sequences: A387008 A387009 A387010 * A387012 A387013 A387014

Keyword

nonn

Author

Seiichi Manyama, Aug 12 2025

Status

approved